On Equation for Initial Values in Theory of the Second Order Ordinary Differential Equations

2002/07/19 by Valerii Dryuma, Valerii S. Dryuma, Dryuma, Valerii S. +3
Mathematics · Physics and Astronomy · #51-XX #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #advanced mathematical theories #msc:51-XX #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0207035

8 pages, Latex

arxiv created 2002/07/19 · openalex publication_date 2002/07/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the properties of the second order nonlinear differential equations b''= g(a,b,b') with the function g(a,b,b'=c) satisfying the following nonlinear partial differential equation \fracd² g_ccda²-g_c\fracdg_ccda-4\fracdg_bcda+ +4g_cg_bc-3g_bg_cc+6g_bb=0, where: (d)/(da)=(∂)/(∂ a)+c (∂)/(∂ b)+ g (∂)/(∂ c). Any equation b''=g(a,b,b') with this condition on function g(a,b,b') has the General Integral F(a,b,x,y)=0 shared with General Integral of the second order ODE's y''=f(x,y,y') with condition (∂⁴ f)/(∂ y'⁴)=0 on function f(x,y,y') or y''+a₁(x,y)y'³+3a₂(x,y)y'²+3a₃(x,y)y'+a₄(x,y)=0 with some coefficients a_i(x,y).

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